4. The McMahon System

Mon, 03/05/2010 - 14:15 — Jon Diamond

4.1 Overview

Most British Go tournaments use the McMahon system, which
is designed to ensure that games in a tournament are most likely to be
even. Each player in the tournament starts off with a McMahon score (or
"MMS") that corresponds to his grade. For example, a 4-kyu player starts
at -4, and a 1-dan player starts at 0 (there is no 0-dan!). Each win for a player
increases his MMS by one, and the winner is the
player with the highest MMS at the end of the tournament.

The draw program (or human organiser) attempts to pair
players with the same MMS against each other. This has the effect that, if a
player enters at the wrong grade, her MMS will gradually come closer to that
of players of her own strength. For example, if a player declares too high a
grade, he is less likely to win, and so his MMS will stay the same while
other players' scores rise – until finally the player meets those of
roughly the same strength.

4.2 The Bar

Because a player's starting score is determined by their grade, a player
who was 7 dan would have a massive advantage and the best chance to win the
tournament, as such a player would start with a very high MMS. To counteract
this, and to give as many people as possible a reasonable chance of winning the
tournament, players at or above a certain rank all begin at the same MMS.
This rank is called the McMahon bar. For example, if the bar is set at 3 dan (which
is an MMS of 2) then no player can start at an MMS of more than 2, no matter
what his or her grade: 3-dans, and all players stronger than 3-dan, also start with an
MMS of 2, and are said to start "above the bar". The position for the bar depends on the number of rounds to be played and also the entry at each of the higher grades. BGA recommended guidelines are as follows:

3 rounds

4-8 players

4 rounds

5-10 players

5 rounds

6-12 players

6 rounds

7-15 players

7 rounds

8-18 players

8 rounds

9-22 players

9 rounds

10-26 players

10 rounds

11-30 players

These figures attempt to meet the following conditions:

There must be a unique winner. This sets an absolute upper limit, of 2r
players above the bar, where r is the number of rounds.

If there are too many above the bar, the tournament will end without all of the top players having played each other.

If there are too few above the bar, these receive an unfair (and unnecessary) disadvantage.

Higher graded players should not run out of even game opponents.

The McMahon System imposes two quite severe constraints on the pairing of players at each round. The first is the rule that there are no repeat games. This increasingly restricts the opponents of the stronger players in the later rounds. The second is the aim of pairing players on the same MMS, which provides the main pairing diversity in the early rounds.

These two pairing rules, together with the nature of the winning probability between players of different grades, provides quite an important ingredient in determining the position of the bar. It turns out that players in the bar group have at least some chance of winning the tournament if the difference between the maximum grade and the bar grade (the bar-depth) is less than 3, whatever the number of rounds [See BGJ 173 Finding The Bar].

If you are using Geoff Kaniuk's GoDraw to create the draw, it will automatically set the bar according to the above table taking into acount the restriction on the bar depth. This is particularly effective in tournaments where the dan entry is very fragmented with possible gaps in the higher grades.

In effect the top players play a Knockout to determine the winner, with losers able to continue at a lower MMS. There
should not be more than 8 players above the bar for a 3 round tournament or 16 players for a 4 round tournament. For higher number of rounds the 2r rule becomes impractical and then we may not get a unique winner based on MMS alone.

(Note: In some larger European tournaments, there is a supergroup, which is
one or more points above the McMahon bar. This is used where there would otherwise be
too many 4-dan players above the McMahon bar.)

4.3 Handicaps

Although the McMahon system decreases the chances of uneven games, they
still occur, especially where there is a large range of entry grades.
The handicap in the McMahon system is normally one less than the current
difference in the players' McMahon scores, with a handicap of 1 meaning
a no-komi game. If there is no handicap, colours are selected (more or
less) at random. Therefore, a player may end up taking White even against
someone on a McMahon score one better than them. It is normal to try to
organise the draw so that, as far as possible, players play an equal
number of games as each colour.

If either player declared a grade at or above the bar, then the
game must be even.

It is not a requirement that the handicap be one less than the McMahon
difference; it is normal in Europe to use difference-2, and some tournaments
use the difference directly. You may specify what you choose. If you
don't want to think about it, specify "McMahon difference minus 1".

4.4 Sleeping Players

A player who misses a round (with the tournament director's consent) sleeps
for that round. For the purpose of producing the pairing, a sleeping player
is deemed to have achieved an average score for each round missed. In order
to prevent biassing the draw by pairing players who sleep, an extra
McMahon point is awarded after every two rounds missed. This score increase
does not count as a win.

In more important tournaments, players in the top McMahon group (or
Supergroup if there is one) are not allowed to sleep for any round. Sleeping
players who would be in the top group are removed by reducing their initial
McMahon score by one or possibly two points. This prevents the lower group
players from interfering with top group players in later rounds.

4.5 Forfeits

If a player wins by default (usually because their opponent fails to show up),
their MMS is increased by one. This counts as a win for the purposes of the
tournament, but not for the purposes of the EGF rating system. It is not counted
as a missed round for either player. (A win by default should not be entered as
a normal win, as we do not want the EGF rating system treating it as a win.)

4.6 Overall winner

At the end of the tournament, the winner is the player with the highest McMahon
score.

There may be a tie for first place (and it is very likely that there will be
ties for other positions). You may consider that this does not matter, and there
is no problem with sharing the first prize. This is your decision. However it
is usual to want a single winner, and to ensure this, a tie-breaking system
should be decided and stated before the tournament. It may consist of just one
of the tie breaks listed below, but the chance of breaking a tis is greater if
several tie breaks are used, in a specified order of precedence.

Some sensible options for a tie-breaking system are list below.
If you don't want to think about it, just specify the first of these.

SOS

(none)

SOS then CUSP

CUSP

Possible tie breaks include:

SOS

(Sum of Opponents' Scores). SOS is the sum over all rounds of the
final McMahon scores of the player's opponents. For each round that the player
sleeps, add the player's own initial McMahon score.

This is the most commonly used tie break in the UK.

CUSS

(Cumulative Sum of Scores). CUSS is the sum over all rounds of
the player's individual McMahon score at the end of each round. This rewards
players who win early in the tournament.

CUSP

(Cumulative Sum of Points). CUSP is the sum over all rounds of
the player's total winning points at the end of each round. Note that winning
points at any round is the total sum of wins and jigos and does not include
free wins or points gained by sleepers.

If two players start off on the same McMahon score, and end up with the
same CUSS (having played all games) then their CUSP will also be the same.
This follows since at any round McMahon score is equal to initial McMahon
score plus winning points up to that round. So one cannot sensibly use both
CUSS and CUSP as first and second level tie-breakers. CUSP is easy to calculate
by hand and is therefore recommended for use as the tie-breaker in small
tournaments.

SoSOS

(Sum of Sum of Opponents Scores). SOSOS is the sum over all
rounds of the SOS scores of the player's opponents. For each round that the
player sleeps add the player's own SOS.

You should decide in advance what to do in case there is still a tie, as even
these tie breakers are not necessarily enough to separate players. For most
tournaments, it is reasonable for first place to be shared. For more important
tournaments, such as the Challenger's League, a play-off game is used.